#### Vol. 301, No. 1, 2019

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Light groups of isomorphisms of Banach spaces and invariant LUR renormings

### Leandro Antunes, Valentin Ferenczi, Sophie Grivaux and Christian Rosendal

Vol. 301 (2019), No. 1, 31–54
##### Abstract

Megrelishvili (2001) defines light groups of isomorphisms of a Banach space as the groups on which the weak and strong operator topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the point of continuity property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

##### Keywords
groups of isomorphisms of Banach spaces, isometry groups, LUR renormings, renormings of Banach spaces, light groups, invariant renormings
##### Mathematical Subject Classification 2010
Primary: 22F50, 46B03
##### Milestones
Received: 21 December 2017
Revised: 25 May 2018
Accepted: 26 September 2018
Published: 16 September 2019
##### Authors
 Leandro Antunes Departamento de Matemática Universidade Tecnológica Federal do Paraná Campus Toledo Toledo, PR Brazil Valentin Ferenczi Departamento de Matematicá Universidade de São Paulo Instituto de Matemática e Estatística São Paulo, SP Brazil Sophie Grivaux CNRS, Univ. Lille, UMR 8524 - Laboratoire Paul Painlevé France Christian Rosendal Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago Chicago, IL United States